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Theorem bj-inf2vnlem3 10910
Description: Lemma for bj-inf2vn 10912. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-inf2vnlem3.bd1  |- BOUNDED  A
bj-inf2vnlem3.bd2  |- BOUNDED  Z
Assertion
Ref Expression
bj-inf2vnlem3  |-  ( A. x  e.  A  (
x  =  (/)  \/  E. y  e.  A  x  =  suc  y )  -> 
(Ind  Z  ->  A  C_  Z ) )
Distinct variable groups:    x, y, A   
x, Z, y

Proof of Theorem bj-inf2vnlem3
Dummy variables  z  t  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-inf2vnlem2 10909 . . 3  |-  ( A. x  e.  A  (
x  =  (/)  \/  E. y  e.  A  x  =  suc  y )  -> 
(Ind  Z  ->  A. u
( A. t  e.  u  ( t  e.  A  ->  t  e.  Z )  ->  (
u  e.  A  ->  u  e.  Z )
) ) )
2 bj-inf2vnlem3.bd1 . . . . . 6  |- BOUNDED  A
32bdeli 10780 . . . . 5  |- BOUNDED  z  e.  A
4 bj-inf2vnlem3.bd2 . . . . . 6  |- BOUNDED  Z
54bdeli 10780 . . . . 5  |- BOUNDED  z  e.  Z
63, 5ax-bdim 10748 . . . 4  |- BOUNDED  ( z  e.  A  ->  z  e.  Z )
7 nfv 1462 . . . 4  |-  F/ z ( t  e.  A  ->  t  e.  Z )
8 nfv 1462 . . . 4  |-  F/ z ( u  e.  A  ->  u  e.  Z )
9 nfv 1462 . . . 4  |-  F/ u
( z  e.  A  ->  z  e.  Z )
10 nfv 1462 . . . 4  |-  F/ u
( t  e.  A  ->  t  e.  Z )
11 eleq1 2142 . . . . . 6  |-  ( z  =  t  ->  (
z  e.  A  <->  t  e.  A ) )
12 eleq1 2142 . . . . . 6  |-  ( z  =  t  ->  (
z  e.  Z  <->  t  e.  Z ) )
1311, 12imbi12d 232 . . . . 5  |-  ( z  =  t  ->  (
( z  e.  A  ->  z  e.  Z )  <-> 
( t  e.  A  ->  t  e.  Z ) ) )
1413biimpd 142 . . . 4  |-  ( z  =  t  ->  (
( z  e.  A  ->  z  e.  Z )  ->  ( t  e.  A  ->  t  e.  Z ) ) )
15 eleq1 2142 . . . . . 6  |-  ( z  =  u  ->  (
z  e.  A  <->  u  e.  A ) )
16 eleq1 2142 . . . . . 6  |-  ( z  =  u  ->  (
z  e.  Z  <->  u  e.  Z ) )
1715, 16imbi12d 232 . . . . 5  |-  ( z  =  u  ->  (
( z  e.  A  ->  z  e.  Z )  <-> 
( u  e.  A  ->  u  e.  Z ) ) )
1817biimprd 156 . . . 4  |-  ( z  =  u  ->  (
( u  e.  A  ->  u  e.  Z )  ->  ( z  e.  A  ->  z  e.  Z ) ) )
196, 7, 8, 9, 10, 14, 18bdsetindis 10907 . . 3  |-  ( A. u ( A. t  e.  u  ( t  e.  A  ->  t  e.  Z )  ->  (
u  e.  A  ->  u  e.  Z )
)  ->  A. z
( z  e.  A  ->  z  e.  Z ) )
201, 19syl6 33 . 2  |-  ( A. x  e.  A  (
x  =  (/)  \/  E. y  e.  A  x  =  suc  y )  -> 
(Ind  Z  ->  A. z
( z  e.  A  ->  z  e.  Z ) ) )
21 dfss2 2989 . 2  |-  ( A 
C_  Z  <->  A. z
( z  e.  A  ->  z  e.  Z ) )
2220, 21syl6ibr 160 1  |-  ( A. x  e.  A  (
x  =  (/)  \/  E. y  e.  A  x  =  suc  y )  -> 
(Ind  Z  ->  A  C_  Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 662   A.wal 1283    = wceq 1285    e. wcel 1434   A.wral 2349   E.wrex 2350    C_ wss 2974   (/)c0 3252   suc csuc 4122  BOUNDED wbdc 10774  Ind wind 10864
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-bdim 10748  ax-bdsetind 10906
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3406  df-suc 4128  df-bdc 10775  df-bj-ind 10865
This theorem is referenced by:  bj-inf2vn  10912
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